I think "sawed-off shotguns" got their reputation the same way the "Tommy Gun" got its reputation: gangsters in the 1920s & 30s. In its heyday, the sawed off shotgun was a formidable weapon. Easily concealed under a coat but devastating (and somewhat indiscriminate) inside, say, a bar.
Everything that's gun related that has a reputation comes from stuff that happened almost 100 years ago. People think that if machine gun registries are opened, it's going to be Chicago Prohibition violence in the streets every day when in reality, between '34 and '86, I can't think of any shootings that involved legal and registered machine guns.
But because the mental imagery of machine guns is Italian mobsters... machine guns are essentially banned.
Sawed off shotguns are subjected to the same mental image suppressors are. Yes, short shotguns were being used for decades in the early 1900's... and times have changed. Handguns that are smaller and hold 10-15 rounds are the choice of robbers and criminals today. Short shotguns... a legal person isn't going to use a short barrel shotgun in a crime, they want it to defend their homes. Criminals, who are already prohibited persons, who have a shotgun aren't going to read the NFA and say, "Oh, it would be a crime for me to hack the barrel down and chop the stock off." They're gonna do it regardless.
At this point, the SBS and SBR laws are used to add a few years to someone's prison sentence. If the goal is to add time to a person who is convicted or pleads to a gun crime, raise the minimum prison sentence. It shouldn't matter if they used an AR, a sawed off shotgun, a musket... if they use a firearm, it should be 15 years minimum with no parole.
Today, 15 rounds of 9mm hollow points are as devastating or more devastating than 5 shots of buckshot out of a 12 gauge. I'd just like to see the barrel length requirement for Short Barrel Shotguns reduced from 18 inches to 10 inches. Keep the law that if it doesn't have a buttstock, it's a sawed off shotgun. I know I'm not going to cut the stock down, that's the part I want most on shotguns.
As for SBR's, that law is even more asinine. I'd just remove that entirely. Handguns will always be more concealable and used in crimes than a rifle with a 15 inch or shorter barrel.